On weak mixing in lattice models

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For lattice models on ℤ d , weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region. For a wide class of models on ℤ2, including all finite range models, we show that weak mixing is a consequence of Gibbs uniqueness, exponential decay of an appropriate form of connectivity, and a natural coupling property. In particular, on ℤ2, the Fortuin-Kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially, and the q-state Potts model above the critical temperature is weak mixing whenever correlations decay exponentially, a hypothesis satisfied if q is sufficiently large. Ratio weak mixing is the property that uniformly over events A and B occurring on subsets Λ and Γ, respectively, of the lattice, |P(A∩B)/P(A)P(B)−1| decreases exponentially in the distance between Λ and Γ. We show that under mild hypotheses, for example finite range, weak mixing implies ratio weak mixing.